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Chapter 5: Problem 14

If \(5 x+4 y=22\) and \(3 x+5 y=21\), then \(x\) is: A. 2.1 B. 2.2 C. 2.5 D. 2.0 E. 2.8

Short Answer

Expert verified
x = 2. Thus, the correct answer is D: 2.0.

Step by step solution

01

- Write down both equations

The given system of equations is: 1) 5x + 4y = 222) 3x + 5y = 21
02

- Eliminate one of the variables

We need to eliminate one variable to solve for the other. Let's eliminate y. To do this, multiply the first equation by 5 and the second equation by 4 to get the coefficients of y to be equal: (5x + 4y) * 5 = 22 * 5 (3x + 5y) * 4 = 21 * 4This gives us: 25x + 20y = 11012x + 20y = 84
03

- Subtract the new equations

Subtract the second new equation from the first new equation to eliminate y: (25x + 20y) - (12x + 20y) = 110 - 84 This simplifies to: 13x = 26
04

- Solve for x

Divide both sides of the equation by 13 to solve for x: x = 26 / 13 Therefore, x = 2
05

- Verify the solution

Substitute x = 2 back into one of the original equations to verify. Using the first equation: 5(2) + 4y = 22 10 + 4y = 22 4y = 12 y = 3 This confirms that our solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

system of linear equations
A system of linear equations consists of two or more equations with the same variables. The objective is to find the values of these variables that satisfy all equations in the system simultaneously. In this exercise, we have a system involving two variables, x and y:
1) \(5x + 4y = 22\)
2) \(3x + 5y = 21\).
These equations need to be solved together to find the values of x and y that make both equations true.
substitution method
The substitution method involves solving one of the equations for one variable and then substituting this expression into the other equation. Although this method works efficiently for many cases, it might not always be the most straightforward method. In the given system, the substitution method could become cumbersome due to the coefficients of x and y.
For instance, if we solve 5x + 4y = 22 for x, we get:
\( x = \frac{22 - 4y}{5} \).
We then substitute this in the second equation:
\( 3(\frac{22 - 4y}{5}) + 5y = 21 \).
This produces a more complex equation compared to using the elimination method.
elimination method
The elimination method involves adding or subtracting the equations to eliminate one of the variables. This strategy is often quicker for systems like the one in this exercise.
Let's eliminate y first by making their coefficients equal. Multiply the first equation by 5 and the second by 4:
\[(5x + 4y) \cdot 5 = 110\]
\[(3x + 5y) \cdot 4 = 84\]
We then get:
\[25x + 20y = 110\]
\[12x + 20y = 84\]
Subtract the second equation from the first:
\[25x + 20y - (12x + 20y) = 110 - 84\]
This leads to:\[13x = 26\]
Solving for x, we find:\[x = 2\]
verification of solutions
Once we have obtained the value of x, it is crucial to verify our solution by substituting back into the original equations. This step ensures our solution is correct. Using our found value of x = 2:
Substitute into the first equation:
\[ 5(2) + 4y = 22 \]
\[ 10 + 4y = 22 \]
\[ 4y = 12 \]
\[ y = 3 \].
Check with the second equation:
\[ 3(2) + 5(3) = 21 \]
\[ 6 + 15 = 21 \].
Both equations are satisfied with x = 2 and y = 3, confirming our solution is correct.

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